aluminium carbonate

Aluminium is a group III element. To form a cation, the size of it would be very small. Therefore there is a very high charge density for Al 3+. This high charge density would disturb the electron cloud of most anion or non-metal.
As carbonate has two negative charge, a very strong electrostatic force would actually break the carbonate ion. So, it does not exist.
Moreover, Al would only form ionic compund with very electronetative elements, usch as F.

aluminium is in group III, the size of cation ion is smaller.

the size of carbonate ion is relatively large.

so it is unstable for them to combine together.